You'll probably need quantum mechanics in Stat Mech before you get enough of it (expecially if QM is 1 year and SM is 1 semester). Check the course syllabus and talk to the prof to see if it will work out.

I was thinking of taking the statistical mechanics course for PhD candidates that's being offered next term at my school. My background is pretty typical, I've had Calc 1-3, linear algebra, differential equations, classical mechanics, e&m, thermodynamics with statistical mechanics, all at the undergraduate level. I'll be taking quantum mechanics concurrently.

You should look at the syllabus. If it's the standard intro graduate stat mech class, it will be undergraduate stat mech again only faster and deeper. If it's advanced stat mech for people that plan to do research in the field, you might have problems, but I think it's more likely it's the first type of course than the second.

The one piece that I think might cause you problems is that you haven't taken quantum yet, and you need some bits of quantum in graduate stat mech. The parts that you'll need are bose-einstein/fermi-dirac statistics and then some practice at how to count quantum states.

I guess I wanted to ask if it was a realistic possibility or will I flounder? Is there any way for me to prepare beforehand to be successful?

Review the bits of quantum that are necessary (Kittel and Kromer's Thermal Physics has a good section on this). Also make sure that you can do PDE's cold as well as the basic thermodynamical calculations.